The dryer outlet is 240V (as was mentioned). There are available switch boxes that allow switching between the dryer and an electric car charger. Only one can operate at a time, but it avoids unplugging the dryer, and it avoids hacking the house wiring. The box has an input cord and two outlet jacks, and a control to select which has power.
Another possibility is to put a 3 phase motor on the lathe and use a 120V to 240V VFD. Since this would be limited to 15 amps it could theoretically develop 2.4 horsepower but generally the available VFDs are limited to 1.0 or 1.5 horsepower at 120V.
Or put a 120v motor on the lathe. Downsize the motor but keep the same RPM. This will reduce cutting performance but may be adequate for most things.